Lattice-ordered matrix algebras over real GCD-domains

Abstract

Let $R \subset \R$ be a GCD-domain. In this paper, Weinberg's conjecture on the $n \times n$ matrix algebra $M_{n}(R) \ (n \geq 2)$ is proved. Moreover, all the lattice orders (up to isomorphisms) on a full $2 \times 2$ matrix algebra over $R$ are obtained.Comment: 9 pages, this is the final version (to appear in the Journal: Communications in Algebra). Thanks to the anonymous referee's helpful suggestions, we replace the UFD rings by the GCD-domains in this pape